Some Nonunique Common Fixed Point Theorems in Symmetric Spaces through CLR (𝑆,𝑇) Property

We introduce a new class of mappings satisfying the “common limit range property” in symmetric spaces and utilize the same to establish common fixed point theorems for such mappings in symmetric spaces. Our results generalize and improve some recent results contained in the literature of metric fixed point theory. Some illustrative examples to highlight the realized improvements are also furnished.


Introduction
In 1986, Jungck [1] generalized the idea of weakly commuting pair of mappings due to Sessa [2] by introducing the notion of compatible pair of mappings and also showed that compatible pair of mappings commute on the set of coincidence points of the involved mappings. Recall that a point ∈ is called a coincidence point of the pair of self-mappings ( , ) defined on if = (= ) while the point is then called a point of coincidence for the pair ( , ). In the recent past and even now, the concept of compatible mappings is frequently used to prove results on the existence of common fixed points. The study of common fixed points of noncompatible pairs is also equally natural and fascinating. Pant [3] initiated the study of noncompatible pairs employing the idea of pointwise -weakly commuting pairs. Pant [4] proved an interesting fixed point theorem for maps satisfying Lipschitz type conditions. In recent years, the result of Pant [4] was generalized and improved by Sastry and Murthy [5] (see also [6]) by introducing the idea of tangential maps (or the property (E.A)) and -continuity. In continuation of this, Imdad and Soliman [7] and Soliman et al. [8] extended the results of Sastry and Murthy [5] as well as Pant [4] to symmetric space utilizing the idea of weakly compatible pair together with common property (E.A) (a notion due to Liu et al. [9]). For more references on the recent development of common fixed point theory in symmetric spaces, we refer readers to [10][11][12][13][14]. Most recently, Gopal et al. [15] improved these results by utilizing the idea of absorbing pair which is essentially due to Gopal et al. [16].
In this paper, we introduce a new notion called the common limit range property and show that this new notion buys a typically required condition up to a pair of mappings along with the notion of absorbing property in proving common fixed point theorems for Lipschitz type mappings in symmetric spaces. Consequently, the relevant recent fixed point theorems due to Soliman et al. [8] and Gopal et al. [15] are generalized and improved.

Preliminaries
A symmetric on a nonempty set is a function : × → [0, ∞) which satisfies ( , ) = ( , ) and ( , ) = 0 ⇔ = (for all , ∈ ). If is a symmetric on a set , then for ∈ and > 0, we write ( , ) = { ∈ : ( , ) < }. A topology ( ) on is given by the sets (along with empty set) in which for each ∈ , one can find some > 0 such that ( , ) ⊂ . A set ⊂ is a neighbourhood of ∈ if and only if there is a containing such that ∈ 2 International Journal of Mathematics and Mathematical Sciences ⊂ . A symmetric is said to be a semimetric if for each ∈ and for each > 0, ( , ) is a neighbourhood of in the topology ( ). Thus a symmetric (resp. a semimetric) space is a topological space whose topology ( ) on is induced by a symmetric (resp. a semimetric) . Notice that lim → ∞ ( , ) = 0 if and only if → in the topology ( ). The distinction between a symmetric and a semimetric is apparent as one can easily construct a semimetric such that ( , ) need not be a neighbourhood of in ( ).
Since symmetric spaces are not necessarily Hausdorff and the symmetric is not generally continuous, in the course of proving fixed point theorems, some additional axioms are required. The following axioms are relevant to this note which are available in the papers of Aliouche [17], Galvin and Shore [18], Hicks and Rhoades [19], and Wilson [20].  (1 ) [18] A symmetric is said to be 1-continuous if lim → ∞ ( , ) = 0 implies lim → ∞ ( , ) = ( , ).
Clearly, ( ) implies (1 ) but not conversely. Also Recall that a sequence { } in a semimetric space ( , ) is said to be -Cauchy if it satisfies the usual metric condition. Here, one needs to notice that in a semimetric space, Cauchy convergence criterion is not a necessary condition for the convergence of a sequence but this criterion becomes a necessary condition if semimetric is suitably restricted (see Wilson [20]). In [22], Burke furnished an illustrative example to show that a convergent sequence in semimetric spaces need not admit a Cauchy subsequence. He was able to formulate an equivalent condition under which every convergent sequence in semimetric space admits a Cauchy subsequence. There are several concept of completeness in semimetric space for example, -completeness, -Cauchy completeness, strong and weak completeness (see Wilson [20]). We omit the details of these notions which are not relevant to this paper.
Let ( , ) be a pair of self-mappings defined on a nonempty set equipped with a symmetric (semimetric) . Then for the pair ( , ), we recall some relevant concepts as follows.
Let be an arbitrary set and be a nonempty set equipped with symmetric (semimetric) . Then the pairs ( , ) and ( , ) of mappings from into are said to have, (iv) (cf. [9]) the common property (E.A) if there exist two sequences { } and { } in such that while the pair ( , ) is said to have (v) the common limit range property with respect to the map (denoted by (CLR ) (cf. [25][26][27][28][29] (vi) let be an arbitrary set and be a nonempty set equipped with symmetric (semimetric) . Then is said to be -continuous (cf. [5]) if → ⇒ → whenever { } is a sequence in and ∈ , (vii) a pair ( , ) of self-mappings defined on a set is said to be weakly compatible (or partially commuting or coincidentally commuting (cf. [5,30])) if the pair commutes on the set of coincidence points that is, = (for ∈ ) implies that = , (viii) let and ( ̸ = ) be two self-mappings defined on a symmetric (or semimetric) space ( , ), then is called -absorbing if there exists some real number > 0 such that ( , ) ≤ ( , ) for all in . Analogously, will be called -absorbing (cf. [16]) if there exists some real number > 0 such that ( , ) ≤ ( , ) for all in . The pair of self maps ( , ) will be called absorbing if it is both -absorbing as well as -absorbing, (ix) let and ( ̸ = ) be two self-mappings defined on a symmetric (or semimetric) space ( , ), then is called pointwise -absorbing if for given in , there exists some > 0 such that ( , ) ≤ ( , ), On similar lines, we can define pointwise -absorbing map. If we take = , the identity map on , then is trivially -absorbing. Similarly, is -absorbing in respect of any . It has been shown in [16] that a pair of compatible or -weakly commuting pair need not be -absorbing or International Journal of Mathematics and Mathematical Sciences 3 -absorbing. Also absorbing pairs are neither a subclass of compatible pairs nor a subclass of noncompatible pairs as the absorbing pairs need not commute at their coincidence points. For other properties and related results for absorbing pair of maps, one can consult [16].
For the sake of completeness, we state below some theorems contained in Soliman et al. [8] and Gopal et al. [15].
Theorem 1 (see cf. [8]). Let be an arbitrary nonempty set while be another nonempty set equipped with a symmetric (semimetric) which enjoys ( 3 ) (Hausdorffness of ( )) and (HE). Let , , , : → be four mappings which satisfy the following conditions: In this paper, we provide a unified approach to certain theorems in symmetric (semimetric) spaces using a blend of common limit range property along with absorbing pair property and obtain generalizations of various results due to Gopal et al. [15], Soliman et al. [8], Pant [31], Sastry and Murthy [5], Imdad et al. [7], Cho et al. [21], and some others.

Main Results
We start to section with the following definition.
If = and = , then the above definition implies (CLR ) property due to Sintunavarat and Kumam [28]. Also notice that the preceeding definition implies the common property (E.A) but the converse implication is not true in general. The following example substantiates this fact.  We now prove our first result employing -continuity of and -continuity of instead of utilizing some Lipschitz or contractive type condition.
which show that ( = ) is a common fixed point of , , and . This concludes the proof.
With a view to demonstrate the utility of Theorem 7 over Theorem 1 and Theorem 2, we adopt the following example.  Notice that at = 1, the involved maps do not satisfy the condition whenever the right hand side is nonzero. Moreover, it can also be verified that at points = 1 and = 2, the involved maps do not satisfy the Lipschitz type condition employed in [4]. Thus, this example substantiates the fact that Theorem 7 is genuine extension of Theorems 1 and 2. By restricting , , , and suitably, one can derive corollaries involving two as well as three mappings. Here, it may be pointed out that any result involving three maps is itself a new result. For the sake of brevity, we opt to mention just one such corollary by restricting Theorem 7 to three mappings , , and which is still new and presents yet another sharpened form of a relevant theorem contained in [15] besides admitting a nonself setting upto coincidence points. The following example illustrates the preceding corollary involving a pair of two self-mappings.
By routine calculations, one can easily verify that the maps in the pair ( , ) satisfies all the conditions of Corollary 9 and have two common fixed points, namely: 2 and 11. Also, the present example does not satisfy the Lipschitz type condition utilized in [4]. To view this claim, consider = 13 and = 22, then we have 9 − 1 ≤ 0 = 0, which is a contradiction. Also, observe that at = 21, the involved maps do not satisfy the condition: whenever the right hand side is nonzero. Here, it is worth noting that none of the earlier relevant theorems for example, Imdad and Soliman [7], Soliman et al. [8] and Gopal et al. [15] can be used in the context of this example as Corollary 9 does not require conditions on containment and closedness amongst the ranges of the involved mappings. Our next theorem is essentially inspired by Theorem 3 due to Gopal et al. [15].

Theorem 11. Let
be an arbitrary set while ( , ) be a symmetric (semimetric) space equipped with a symmetric (semimetric) which enjoys ( 3 ) (or Hausdorffness of ( )) and ( ). If , , , : → are four mappings which satisfy the following conditions: which show that ( = ) is a common fixed point of , , , and .
The following example demonstrates Theorem 11.
Then, by a routine calculation, it can be easily verified that and satisfy condition (ii) (of Theorem 11) for = 4.271. Also, the mappings and satisfies the CLR ( , ) property with the sequence = 5 + 1/ . The verification of the pointwise absorbing property of the pair ( , ) is straight forward. Thus and satisfy all the conditions of Theorem 11 and have two common fixed points, namely: = 2 and = 10.
Remark 13. Choosing = 1 in Theorem 11, we can derive a slightly sharpened form of a theorem due to Cho et al. [21] as conditions on the ranges of involved mappings are completely relaxed.
By restricting , , , and suitably, one can derive corollaries for two as well as three mappings. For the sake of brevity, we derive just one corollary by restricting Theorem 11 to three mappings which is yet another sharpened and unified form of a theorem due to Gopal et al. [15] in symmetric spaces and also remains relevant to some results in Pant [4] and Pant [31].  Proof. Proof follows from Theorem 11 by setting = 1.
Our next theorem is essentially inspired by a Lipschitzian condition utilized by Cho et al. [21] as well as Gopal et al. [15]. Proof. The proof can be completed on the lines of proof of Theorem 11, hence details are not included.
By restricting , , , and suitably, one can derive corollaries for two as well as three mappings. For the sake of brevity, we derive just one corollary by restricting Theorem 16 to three mappings which is yet another sharpened form of a theorem contained in [15] which also remains relevant to some results in Pant [4] and Pant [31]. Proof. The proof can be completed on the lines of proof of Theorem 11.